Dear Allstat Community,<div><br></div><div>Thank you to those who responded to my first email to ask for help.  I realize that my blog post is a little long to describe my problem, and I could have included a few key details in my earlier email to make my questions more clear.  I apologize for any confusion, and I am writing again to clarify the context of my problem.  I thank you for your patience in advance.</div>



<div><br></div><div><br></div><div>- The goal is to estimate the decay rate and half-life in an exponential decay model, which is</div><div><br></div><div>C(Year) = C0*exp(-lambda*Year)</div><div><br></div><div>where C(Year) is the DDT concentration for a given year and C0 is the concentration of DDT in the first year.  </div>



<div><br></div><div><br>- Data were collected on DDT concentration over several years.  Please see my blog post (<a href="http://wp.me/p3bFTa-4R" target="_blank">http://wp.me/p3bFTa-4R</a>) for the data set.  </div><div>


<br></div><div><br></div><div>- The above model of exponential decay is clearly not linear between C(Year) and Year, so it was log-transformed into </div>
<div><br></div><div>ln[C(Year)] = ln[C0] - lambda*Year</div><div><br></div><div><br clear="all"><div><div>- The log-transformed data shows linearity, but the residuals-vs-fitted and residuals-vs-year plots show moderately decreasing variance</div>



<div><br></div><div><br></div><div><br></div><div>Here are my difficulties again; I would most appreciate any ideas on how to overcome these problems.</div><div><br></div><div>1) After the logarithmic transformation to linearize the exponential decay relationship, I&#39;m still getting decreasing variance.  I can&#39;t find a suitable transformation on top of this logarithmic transformation to make the variance constant.  Square root or another logarithmic transformation is obviously undefined for negative values, and any exponentiation would just reverse the logarithmic linearization.</div>


<div><br></div><div>2) I estimated the half-life (see end of my blog post: <a href="http://wp.me/p3bFTa-4R" target="_blank">http://wp.me/p3bFTa-4R</a>), but I&#39;m struggling to calculate a confidence interval for it, since the calculation of its variance involves the reciprocal of the estimated decay rate (lambda).</div>


<div><br></div><div>Var[ln(0.5)/lambda] = Var[Half-life]</div><div><br></div><div><br></div><div><br></div><div>Thanks again for your thoughts.</div><div><br></div><div><br></div><div>Eric</div><div><br></div><div>________________________</div>


<div>Eric Cai</div><div>The Chemical Statistician</div><div><a href="http://chemicalstatistician.wordpress.com/" target="_blank">http://chemicalstatistician.wordpress.com/</a></div>
<div><br></div><div>Twitter: @chemstateric</div><div><a href="https://twitter.com/chemstateric" target="_blank">https://twitter.com/chemstateric</a></div><div><br></div><div>M.Sc.</div><div>Statistics</div><div>University of Toronto</div>



<div><br></div><div>B.Sc. with Distinction</div><div>Chemistry and Mathematics</div><div>Simon Fraser University</div></div>
<div><br></div><br><br><div class="gmail_quote">On Tue, Mar 26, 2013 at 9:46 AM, Eric Cai <span dir="ltr">&lt;<a href="mailto:[log in to unmask]" target="_blank"><a href="/cgi-bin/wa-jisc.exe?LOGON=A3%3Dind1303%26L%3DALLSTAT%26E%3Dquoted-printable%26P%3D3007520%26B%3D--089e0158b3aac949f204d8e72e53%26T%3Dtext%252Fhtml%3B%2520charset%3DISO-8859-1%26pending%3D" target="_parent" >[log in to unmask]</a></a>&gt;</span> wrote:<br>


<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>Dear Allstat Community,</div><div><br></div><div>In a recent post on my blog, I demonstrated how logarithmic transformation can linearize a target-predictor relationship. </div><div><br></div><div><a href="http://wp.me/p3bFTa-4R" target="_blank">http://wp.me/p3bFTa-4R</a></div>





<div><br></div><div><br></div><div>I would much appreciate your help with 2 sets of difficulties.</div><div><br></div><div>1) If you plot the residuals against the fitted values of ln(DDT), you will find that the variance decreases. I am encountering 2 problems:</div>





<div>1a) I can&#39;t find a good transformation to remove this trend in the variance.</div><div><br></div><div>1b) I have already transformed the model once. I fear that transforming it again would make it difficult to make inferences on the quantities and obtain an easily understandable interpretation of the quantities.  </div>





<div><br></div><div><br></div><div><br></div><div>2) At the end of the blog post, I used the estimated decay rate to calculate the half-life. I would like to calculate a confidence interval for the half-life. Even if I (incorrectly) set aside the non-constant variance that ruins my estimation of the standard errors, I don&#39;t know how to calculate a confidence interval for the half-life, since the variance calculation would start off as</div>





<div><br></div><div><br></div><div>Var[ln(0.5)/lambda] = Var[half-life],</div><div><br></div><div><br></div><div>where lambda is the random variable in this case; its standard error is given in my regression output for beta1. Half-life is not a linear combination of lambda, so the variance cannot be nicely calculated with the usual rules of variance. Thus, I don&#39;t know how to calculate the variance of half-life and, hence, the confidence interval for half-life. (I fully understand that the standard error for beta1/lambda is not good, so I welcome any suggestion for overcoming both of these problems simultaneously.)</div>





<div><br></div><div><br></div><div>I would much appreciate your insights on how I can overcome these problems.</div><div><br></div><div><br></div><div>Thank you for your time.</div><div><br></div><div><br></div><div>Eric</div>





<div><br></div><div><br></div><div>________________________</div><div>Eric Cai</div><div>The Chemical Statistician</div><div><a href="http://chemicalstatistician.wordpress.com/" target="_blank">http://chemicalstatistician.wordpress.com/</a></div>





<div><br></div><div>Twitter: @chemstateric</div><div><a href="https://twitter.com/chemstateric" target="_blank">https://twitter.com/chemstateric</a></div><div><br></div><div>M.Sc.</div><div>Statistics</div><div>University of Toronto</div>





<div><br></div><div>B.Sc. with Distinction</div><div>Chemistry and Mathematics</div><div>Simon Fraser University</div><div><br></div><div><br></div>
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